LMFDB - Embedded newform 576.4.k.a.433.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,4,Mod(145,576)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("576.145");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	576.k (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$33.9851001633$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - x^{8} + 6x^{7} + 14x^{6} - 80x^{5} + 56x^{4} + 96x^{3} - 64x^{2} - 512x + 1024 $ x^10 - 2x^9 - x^8 + 6x^7 + 14x^6 - 80x^5 + 56x^4 + 96x^3 - 64x^2 - 512x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			433.1
Root			$1.97476 - 0.316760i$ of defining polynomial
Character	$\chi$	$=$	576.433
Dual form			576.4.k.a.145.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.67959 - 8.67959i) q^{5} -1.63924i q^{7} +O(q^{10})$	$q+(-8.67959 - 8.67959i) q^{5} -1.63924i q^{7} +(18.2021 + 18.2021i) q^{11} +(-9.34700 + 9.34700i) q^{13} -53.6113 q^{17} +(-70.9870 + 70.9870i) q^{19} -25.1189i q^{23} +25.6706i q^{25} +(181.094 - 181.094i) q^{29} -132.684 q^{31} +(-14.2280 + 14.2280i) q^{35} +(174.872 + 174.872i) q^{37} +198.660i q^{41} +(285.717 + 285.717i) q^{43} +78.3629 q^{47} +340.313 q^{49} +(525.776 + 525.776i) q^{53} -315.973i q^{55} +(46.5301 + 46.5301i) q^{59} +(193.318 - 193.318i) q^{61} +162.256 q^{65} +(-282.182 + 282.182i) q^{67} +727.536i q^{71} -106.065i q^{73} +(29.8376 - 29.8376i) q^{77} +58.9970 q^{79} +(-410.156 + 410.156i) q^{83} +(465.324 + 465.324i) q^{85} +768.959i q^{89} +(15.3220 + 15.3220i) q^{91} +1232.28 q^{95} -809.953 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{5}+O(q^{10}) $ 10 * q + 2 * q^5	$ 10 q + 2 q^{5} + 18 q^{11} - 2 q^{13} + 4 q^{17} + 26 q^{19} + 202 q^{29} - 368 q^{31} + 476 q^{35} - 10 q^{37} + 838 q^{43} - 944 q^{47} + 94 q^{49} + 378 q^{53} + 1706 q^{59} + 910 q^{61} + 492 q^{65} - 1942 q^{67} + 268 q^{77} + 4416 q^{79} - 2562 q^{83} - 12 q^{85} - 3332 q^{91} + 6900 q^{95} - 4 q^{97}+O(q^{100}) $ 10 * q + 2 * q^5 + 18 * q^11 - 2 * q^13 + 4 * q^17 + 26 * q^19 + 202 * q^29 - 368 * q^31 + 476 * q^35 - 10 * q^37 + 838 * q^43 - 944 * q^47 + 94 * q^49 + 378 * q^53 + 1706 * q^59 + 910 * q^61 + 492 * q^65 - 1942 * q^67 + 268 * q^77 + 4416 * q^79 - 2562 * q^83 - 12 * q^85 - 3332 * q^91 + 6900 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−8.67959	−	8.67959i	−0.776326	−	0.776326i	0.202878	−	0.979204i	$-0.434971\pi$
$5$	−8.67959	−	8.67959i	−0.776326	−	0.776326i	−0.979204	+	0.202878i	$0.934971\pi$
$6$		0			0
$7$		−	1.63924i		−	0.0885109i	−0.999020	−	0.0442554i	$-0.985908\pi$
$7$		−	1.63924i		−	0.0885109i	0.999020	−	0.0442554i	$-0.0140915\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	18.2021	+	18.2021i	0.498921	+	0.498921i	0.911102	−	0.412181i	$-0.135233\pi$
$11$	18.2021	+	18.2021i	0.498921	+	0.498921i	−0.412181	+	0.911102i	$0.635233\pi$
$12$		0			0
$13$	−9.34700	+	9.34700i	−0.199415	+	0.199415i	−0.799749	−	0.600334i	$-0.795034\pi$
$13$	−9.34700	+	9.34700i	−0.199415	+	0.199415i	0.600334	+	0.799749i	$0.295034\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−53.6113			−0.764862			−0.382431	−	0.923984i	$-0.624913\pi$
$17$	−53.6113			−0.764862			−0.382431	+	0.923984i	$0.624913\pi$
$18$		0			0
$19$	−70.9870	+	70.9870i	−0.857133	+	0.857133i	−0.990999	−	0.133866i	$-0.957261\pi$
$19$	−70.9870	+	70.9870i	−0.857133	+	0.857133i	0.133866	+	0.990999i	$0.457261\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	25.1189i		−	0.227724i	−0.993497	−	0.113862i	$-0.963678\pi$
$23$		−	25.1189i		−	0.227724i	0.993497	−	0.113862i	$-0.0363222\pi$
$24$		0			0
$25$			25.6706i			0.205365i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	181.094	−	181.094i	1.15960	−	1.15960i	0.175033	−	0.984563i	$-0.443997\pi$
$29$	181.094	−	181.094i	1.15960	−	1.15960i	0.984563	−	0.175033i	$-0.0560031\pi$
$30$		0			0
$31$	−132.684			−0.768733			−0.384367	−	0.923180i	$-0.625580\pi$
$31$	−132.684			−0.768733			−0.384367	+	0.923180i	$0.625580\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−14.2280	+	14.2280i	−0.0687133	+	0.0687133i
$36$		0			0
$37$	174.872	+	174.872i	0.776994	+	0.776994i	0.979319	−	0.202324i	$-0.0648495\pi$
$37$	174.872	+	174.872i	0.776994	+	0.776994i	−0.202324	+	0.979319i	$0.564849\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			198.660i			0.756720i	0.925658	+	0.378360i	$0.123512\pi$
$41$			198.660i			0.756720i	−0.925658	+	0.378360i	$0.876488\pi$
$42$		0			0
$43$	285.717	+	285.717i	1.01329	+	1.01329i	0.999911	+	0.0133770i	$0.00425815\pi$
$43$	285.717	+	285.717i	1.01329	+	1.01329i	0.0133770	+	0.999911i	$0.495742\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	78.3629			0.243200			0.121600	−	0.992579i	$-0.461197\pi$
$47$	78.3629			0.243200			0.121600	+	0.992579i	$0.461197\pi$
$48$		0			0
$49$	340.313			0.992166
$50$		0			0
$51$		0			0
$52$		0			0
$53$	525.776	+	525.776i	1.36266	+	1.36266i	0.870515	+	0.492143i	$0.163786\pi$
$53$	525.776	+	525.776i	1.36266	+	1.36266i	0.492143	+	0.870515i	$0.336214\pi$
$54$		0			0
$55$		−	315.973i		−	0.774650i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	46.5301	+	46.5301i	0.102673	+	0.102673i	0.756577	−	0.653904i	$-0.226870\pi$
$59$	46.5301	+	46.5301i	0.102673	+	0.102673i	−0.653904	+	0.756577i	$0.726870\pi$
$60$		0			0
$61$	193.318	−	193.318i	0.405767	−	0.405767i	−0.474493	−	0.880259i	$-0.657368\pi$
$61$	193.318	−	193.318i	0.405767	−	0.405767i	0.880259	+	0.474493i	$0.157368\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	162.256			0.309622
$66$		0			0
$67$	−282.182	+	282.182i	−0.514538	+	0.514538i	−0.915914	−	0.401375i	$-0.868532\pi$
$67$	−282.182	+	282.182i	−0.514538	+	0.514538i	0.401375	+	0.915914i	$0.368532\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			727.536i			1.21609i	0.793901	+	0.608046i	$0.208047\pi$
$71$			727.536i			1.21609i	−0.793901	+	0.608046i	$0.791953\pi$
$72$		0			0
$73$		−	106.065i		−	0.170054i	−0.996379	−	0.0850270i	$-0.972902\pi$
$73$		−	106.065i		−	0.170054i	0.996379	−	0.0850270i	$-0.0270977\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	29.8376	−	29.8376i	0.0441599	−	0.0441599i
$78$		0			0
$79$	58.9970			0.0840213			0.0420107	−	0.999117i	$-0.486624\pi$
$79$	58.9970			0.0840213			0.0420107	+	0.999117i	$0.486624\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−410.156	+	410.156i	−0.542416	+	0.542416i	−0.924236	−	0.381821i	$-0.875297\pi$
$83$	−410.156	+	410.156i	−0.542416	+	0.542416i	0.381821	+	0.924236i	$0.375297\pi$
$84$		0			0
$85$	465.324	+	465.324i	0.593783	+	0.593783i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			768.959i			0.915837i	0.888994	+	0.457918i	$0.151405\pi$
$89$			768.959i			0.915837i	−0.888994	+	0.457918i	$0.848595\pi$
$90$		0			0
$91$	15.3220	+	15.3220i	0.0176504	+	0.0176504i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1232.28			1.33083
$96$		0			0
$97$	−809.953			−0.847817			−0.423908	−	0.905705i	$-0.639342\pi$
$97$	−809.953			−0.847817			−0.423908	+	0.905705i	$0.639342\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	303.189	+	303.189i	0.298698	+	0.298698i	0.840504	−	0.541806i	$-0.182259\pi$
$101$	303.189	+	303.189i	0.298698	+	0.298698i	−0.541806	+	0.840504i	$0.682259\pi$
$102$		0			0
$103$		−	962.201i		−	0.920471i	−0.887797	−	0.460235i	$-0.847765\pi$
$103$		−	962.201i		−	0.920471i	0.887797	−	0.460235i	$-0.152235\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	728.337	+	728.337i	0.658046	+	0.658046i	0.954918	−	0.296871i	$-0.0959432\pi$
$107$	728.337	+	728.337i	0.658046	+	0.658046i	−0.296871	+	0.954918i	$0.595943\pi$
$108$		0			0
$109$	−593.258	+	593.258i	−0.521319	+	0.521319i	−0.917970	−	0.396651i	$-0.870172\pi$
$109$	−593.258	+	593.258i	−0.521319	+	0.521319i	0.396651	+	0.917970i	$0.370172\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−351.938			−0.292987			−0.146493	−	0.989212i	$-0.546799\pi$
$113$	−351.938			−0.292987			−0.146493	+	0.989212i	$0.546799\pi$
$114$		0			0
$115$	−218.022	+	218.022i	−0.176788	+	0.176788i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			87.8821i			0.0676986i
$120$		0			0
$121$		−	668.370i		−	0.502157i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−862.139	+	862.139i	−0.616896	+	0.616896i
$126$		0			0
$127$	2365.81			1.65301			0.826504	−	0.562931i	$-0.190326\pi$
$127$	2365.81			1.65301			0.826504	+	0.562931i	$0.190326\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	403.454	−	403.454i	0.269083	−	0.269083i	−0.559647	−	0.828731i	$-0.689063\pi$
$131$	403.454	−	403.454i	0.269083	−	0.269083i	0.828731	+	0.559647i	$0.189063\pi$
$132$		0			0
$133$	116.365	+	116.365i	0.0758656	+	0.0758656i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	856.850i		−	0.534348i	−0.963648	−	0.267174i	$-0.913910\pi$
$137$		−	856.850i		−	0.534348i	0.963648	−	0.267174i	$-0.0860898\pi$
$138$		0			0
$139$	1689.33	+	1689.33i	1.03084	+	1.03084i	0.999509	+	0.0313345i	$0.00997572\pi$
$139$	1689.33	+	1689.33i	1.03084	+	1.03084i	0.0313345	+	0.999509i	$0.490024\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−340.269			−0.198984
$144$		0			0
$145$	−3143.64			−1.80045
$146$		0			0
$147$		0			0
$148$		0			0
$149$	32.2208	+	32.2208i	0.0177156	+	0.0177156i	0.715909	−	0.698193i	$-0.246012\pi$
$149$	32.2208	+	32.2208i	0.0177156	+	0.0177156i	−0.698193	+	0.715909i	$0.746012\pi$
$150$		0			0
$151$			1077.06i			0.580460i	0.956957	+	0.290230i	$0.0937319\pi$
$151$			1077.06i			0.580460i	−0.956957	+	0.290230i	$0.906268\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1151.64	+	1151.64i	0.596788	+	0.596788i
$156$		0			0
$157$	1905.61	−	1905.61i	0.968691	−	0.968691i	−0.0308332	−	0.999525i	$-0.509816\pi$
$157$	1905.61	−	1905.61i	0.968691	−	0.968691i	0.999525	+	0.0308332i	$0.00981607\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−41.1761			−0.0201561
$162$		0			0
$163$	−709.828	+	709.828i	−0.341092	+	0.341092i	−0.856778	−	0.515686i	$-0.827537\pi$
$163$	−709.828	+	709.828i	−0.341092	+	0.341092i	0.515686	+	0.856778i	$0.327537\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	3460.66i		−	1.60356i	−0.597623	−	0.801778i	$-0.703888\pi$
$167$		−	3460.66i		−	1.60356i	0.597623	−	0.801778i	$-0.296112\pi$
$168$		0			0
$169$			2022.27i			0.920467i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1303.54	−	1303.54i	0.572871	−	0.572871i	−0.360059	−	0.932930i	$-0.617243\pi$
$173$	1303.54	−	1303.54i	0.572871	−	0.572871i	0.932930	+	0.360059i	$0.117243\pi$
$174$		0			0
$175$	42.0803			0.0181770
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1082.35	+	1082.35i	−0.451947	+	0.451947i	−0.896000	−	0.444054i	$-0.853540\pi$
$179$	−1082.35	+	1082.35i	−0.451947	+	0.451947i	0.444054	+	0.896000i	$0.353540\pi$
$180$		0			0
$181$	2943.93	+	2943.93i	1.20895	+	1.20895i	0.971366	+	0.237588i	$0.0763567\pi$
$181$	2943.93	+	2943.93i	1.20895	+	1.20895i	0.237588	+	0.971366i	$0.423643\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	3035.64i		−	1.20640i
$186$		0			0
$187$	−975.837	−	975.837i	−0.381606	−	0.381606i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−430.650			−0.163145			−0.0815726	−	0.996667i	$-0.525994\pi$
$191$	−430.650			−0.163145			−0.0815726	+	0.996667i	$0.525994\pi$
$192$		0			0
$193$	−2266.98			−0.845497			−0.422749	−	0.906247i	$-0.638935\pi$
$193$	−2266.98			−0.845497			−0.422749	+	0.906247i	$0.638935\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1039.41	+	1039.41i	0.375913	+	0.375913i	0.869625	−	0.493712i	$-0.164360\pi$
$197$	1039.41	+	1039.41i	0.375913	+	0.375913i	−0.493712	+	0.869625i	$0.664360\pi$
$198$		0			0
$199$		−	4989.44i		−	1.77735i	−0.458540	−	0.888674i	$-0.651627\pi$
$199$		−	4989.44i		−	1.77735i	0.458540	−	0.888674i	$-0.348373\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−296.857	−	296.857i	−0.102637	−	0.102637i
$204$		0			0
$205$	1724.29	−	1724.29i	0.587462	−	0.587462i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2584.22			−0.855283
$210$		0			0
$211$	−2651.50	+	2651.50i	−0.865103	+	0.865103i	−0.991925	−	0.126823i	$-0.959522\pi$
$211$	−2651.50	+	2651.50i	−0.865103	+	0.865103i	0.126823	+	0.991925i	$0.459522\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	4959.80i		−	1.57328i
$216$		0			0
$217$			217.501i			0.0680413i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	501.105	−	501.105i	0.152525	−	0.152525i
$222$		0			0
$223$	−3690.85			−1.10833			−0.554165	−	0.832407i	$-0.686962\pi$
$223$	−3690.85			−1.10833			−0.554165	+	0.832407i	$0.686962\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1710.42	+	1710.42i	−0.500108	+	0.500108i	−0.911471	−	0.411363i	$-0.865053\pi$
$227$	−1710.42	+	1710.42i	−0.500108	+	0.500108i	0.411363	+	0.911471i	$0.365053\pi$
$228$		0			0
$229$	−91.8012	−	91.8012i	−0.0264908	−	0.0264908i	0.693737	−	0.720228i	$-0.255963\pi$
$229$	−91.8012	−	91.8012i	−0.0264908	−	0.0264908i	−0.720228	+	0.693737i	$0.755963\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			4259.71i			1.19769i	0.800863	+	0.598847i	$0.204374\pi$
$233$			4259.71i			1.19769i	−0.800863	+	0.598847i	$0.795626\pi$
$234$		0			0
$235$	−680.158	−	680.158i	−0.188803	−	0.188803i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5053.12			−1.36761			−0.683806	−	0.729664i	$-0.739676\pi$
$239$	−5053.12			−1.36761			−0.683806	+	0.729664i	$0.739676\pi$
$240$		0			0
$241$	48.8379			0.0130536			0.00652681	−	0.999979i	$-0.497922\pi$
$241$	48.8379			0.0130536			0.00652681	+	0.999979i	$0.497922\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2953.78	−	2953.78i	−0.770244	−	0.770244i
$246$		0			0
$247$		−	1327.03i		−	0.341850i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2604.76	−	2604.76i	−0.655025	−	0.655025i	0.299174	−	0.954199i	$-0.403289\pi$
$251$	−2604.76	−	2604.76i	−0.655025	−	0.655025i	−0.954199	+	0.299174i	$0.903289\pi$
$252$		0			0
$253$	457.216	−	457.216i	0.113616	−	0.113616i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−739.054			−0.179381			−0.0896905	−	0.995970i	$-0.528588\pi$
$257$	−739.054			−0.179381			−0.0896905	+	0.995970i	$0.528588\pi$
$258$		0			0
$259$	286.658	−	286.658i	0.0687724	−	0.0687724i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			2448.30i			0.574025i	0.957927	+	0.287012i	$0.0926621\pi$
$263$			2448.30i			0.574025i	−0.957927	+	0.287012i	$0.907338\pi$
$264$		0			0
$265$		−	9127.03i		−	2.11573i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	829.952	−	829.952i	0.188116	−	0.188116i	−0.606765	−	0.794881i	$-0.707533\pi$
$269$	829.952	−	829.952i	0.188116	−	0.188116i	0.794881	+	0.606765i	$0.207533\pi$
$270$		0			0
$271$	−1404.85			−0.314902			−0.157451	−	0.987527i	$-0.550328\pi$
$271$	−1404.85			−0.314902			−0.157451	+	0.987527i	$0.550328\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−467.257	+	467.257i	−0.102461	+	0.102461i
$276$		0			0
$277$	−2245.69	−	2245.69i	−0.487112	−	0.487112i	0.420281	−	0.907394i	$-0.361931\pi$
$277$	−2245.69	−	2245.69i	−0.487112	−	0.487112i	−0.907394	+	0.420281i	$0.861931\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	6045.97i		−	1.28353i	−0.766900	−	0.641766i	$-0.778202\pi$
$281$		−	6045.97i		−	1.28353i	0.766900	−	0.641766i	$-0.221798\pi$
$282$		0			0
$283$	−2459.63	−	2459.63i	−0.516643	−	0.516643i	0.399911	−	0.916554i	$-0.369041\pi$
$283$	−2459.63	−	2459.63i	−0.516643	−	0.516643i	−0.916554	+	0.399911i	$0.869041\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	325.653			0.0669780
$288$		0			0
$289$	−2038.82			−0.414986
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1852.49	+	1852.49i	0.369364	+	0.369364i	0.867245	−	0.497881i	$-0.165888\pi$
$293$	1852.49	+	1852.49i	0.369364	+	0.369364i	−0.497881	+	0.867245i	$0.665888\pi$
$294$		0			0
$295$		−	807.725i		−	0.159415i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	234.787	+	234.787i	0.0454116	+	0.0454116i
$300$		0			0
$301$	468.359	−	468.359i	0.0896870	−	0.0896870i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3355.83			−0.630015
$306$		0			0
$307$	−2107.35	+	2107.35i	−0.391768	+	0.391768i	−0.875317	−	0.483549i	$-0.839347\pi$
$307$	−2107.35	+	2107.35i	−0.391768	+	0.391768i	0.483549	+	0.875317i	$0.339347\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			5294.90i			0.965422i	0.875780	+	0.482711i	$0.160348\pi$
$311$			5294.90i			0.965422i	−0.875780	+	0.482711i	$0.839652\pi$
$312$		0			0
$313$			4005.87i			0.723403i	0.932294	+	0.361702i	$0.117804\pi$
$313$			4005.87i			0.723403i	−0.932294	+	0.361702i	$0.882196\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−809.240	+	809.240i	−0.143380	+	0.143380i	−0.775153	−	0.631773i	$-0.782327\pi$
$317$	−809.240	+	809.240i	−0.143380	+	0.143380i	0.631773	+	0.775153i	$0.282327\pi$
$318$		0			0
$319$	6592.56			1.15709
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3805.71	−	3805.71i	0.655589	−	0.655589i
$324$		0			0
$325$	−239.943	−	239.943i	−0.0409527	−	0.0409527i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	128.456i		−	0.0215259i
$330$		0			0
$331$	4229.66	+	4229.66i	0.702366	+	0.702366i	0.964918	−	0.262552i	$-0.0845641\pi$
$331$	4229.66	+	4229.66i	0.702366	+	0.702366i	−0.262552	+	0.964918i	$0.584564\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4898.46			0.798899
$336$		0			0
$337$	10002.6			1.61684			0.808419	−	0.588607i	$-0.200323\pi$
$337$	10002.6			1.61684			0.808419	+	0.588607i	$0.200323\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2415.12	−	2415.12i	−0.383537	−	0.383537i
$342$		0			0
$343$		−	1120.12i		−	0.176328i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6409.49	−	6409.49i	−0.991583	−	0.991583i	0.00838198	−	0.999965i	$-0.497332\pi$
$347$	−6409.49	−	6409.49i	−0.991583	−	0.991583i	−0.999965	+	0.00838198i	$0.997332\pi$
$348$		0			0
$349$	−5503.23	+	5503.23i	−0.844071	+	0.844071i	−0.989386	−	0.145314i	$-0.953581\pi$
$349$	−5503.23	+	5503.23i	−0.844071	+	0.844071i	0.145314	+	0.989386i	$0.453581\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1411.35			0.212800			0.106400	−	0.994323i	$-0.466068\pi$
$353$	1411.35			0.212800			0.106400	+	0.994323i	$0.466068\pi$
$354$		0			0
$355$	6314.71	−	6314.71i	0.944085	−	0.944085i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	2160.73i		−	0.317658i	−0.987306	−	0.158829i	$-0.949228\pi$
$359$		−	2160.73i		−	0.317658i	0.987306	−	0.158829i	$-0.0507718\pi$
$360$		0			0
$361$		−	3219.31i		−	0.469355i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−920.599	+	920.599i	−0.132017	+	0.132017i
$366$		0			0
$367$	−10757.7			−1.53010			−0.765052	−	0.643969i	$-0.777287\pi$
$367$	−10757.7			−1.53010			−0.765052	+	0.643969i	$0.777287\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	861.875	−	861.875i	0.120610	−	0.120610i
$372$		0			0
$373$	1406.99	+	1406.99i	0.195312	+	0.195312i	0.797987	−	0.602675i	$-0.205898\pi$
$373$	1406.99	+	1406.99i	0.195312	+	0.195312i	−0.602675	+	0.797987i	$0.705898\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			3385.37i			0.462481i
$378$		0			0
$379$	1146.95	+	1146.95i	0.155449	+	0.155449i	0.780547	−	0.625098i	$-0.214941\pi$
$379$	1146.95	+	1146.95i	0.155449	+	0.155449i	−0.625098	+	0.780547i	$0.714941\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9042.17			1.20635			0.603176	−	0.797608i	$-0.293902\pi$
$383$	9042.17			1.20635			0.603176	+	0.797608i	$0.293902\pi$
$384$		0			0
$385$	−517.957			−0.0685650
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2575.34	−	2575.34i	−0.335668	−	0.335668i	0.519066	−	0.854734i	$-0.326280\pi$
$389$	−2575.34	−	2575.34i	−0.335668	−	0.335668i	−0.854734	+	0.519066i	$0.826280\pi$
$390$		0			0
$391$			1346.66i			0.174178i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−512.070	−	512.070i	−0.0652279	−	0.0652279i
$396$		0			0
$397$	−7121.46	+	7121.46i	−0.900292	+	0.900292i	−0.995461	−	0.0951695i	$-0.969661\pi$
$397$	−7121.46	+	7121.46i	−0.900292	+	0.900292i	0.0951695	+	0.995461i	$0.469661\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3025.14			0.376729			0.188365	−	0.982099i	$-0.439681\pi$
$401$	3025.14			0.376729			0.188365	+	0.982099i	$0.439681\pi$
$402$		0			0
$403$	1240.20	−	1240.20i	0.153297	−	0.153297i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			6366.06i			0.775317i
$408$		0			0
$409$			9440.21i			1.14129i	0.821196	+	0.570646i	$0.193307\pi$
$409$			9440.21i			1.14129i	−0.821196	+	0.570646i	$0.806693\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	76.2743	−	76.2743i	0.00908768	−	0.00908768i
$414$		0			0
$415$	7119.98			0.842183
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3255.69	+	3255.69i	−0.379597	+	0.379597i	−0.870957	−	0.491360i	$-0.836500\pi$
$419$	−3255.69	+	3255.69i	−0.379597	+	0.379597i	0.491360	+	0.870957i	$0.336500\pi$
$420$		0			0
$421$	−9438.04	−	9438.04i	−1.09259	−	1.09259i	−0.995251	−	0.0973423i	$-0.968966\pi$
$421$	−9438.04	−	9438.04i	−1.09259	−	1.09259i	−0.0973423	−	0.995251i	$-0.531034\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	1376.23i		−	0.157076i
$426$		0			0
$427$	−316.895	−	316.895i	−0.0359148	−	0.0359148i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−10617.7			−1.18663			−0.593314	−	0.804971i	$-0.702181\pi$
$431$	−10617.7			−1.18663			−0.593314	+	0.804971i	$0.702181\pi$
$432$		0			0
$433$	706.479			0.0784093			0.0392046	−	0.999231i	$-0.487518\pi$
$433$	706.479			0.0784093			0.0392046	+	0.999231i	$0.487518\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1783.12	+	1783.12i	0.195190	+	0.195190i
$438$		0			0
$439$			13611.8i			1.47985i	0.672688	+	0.739926i	$0.265140\pi$
$439$			13611.8i			1.47985i	−0.672688	+	0.739926i	$0.734860\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3126.97	+	3126.97i	0.335366	+	0.335366i	0.854620	−	0.519254i	$-0.173790\pi$
$443$	3126.97	+	3126.97i	0.335366	+	0.335366i	−0.519254	+	0.854620i	$0.673790\pi$
$444$		0			0
$445$	6674.25	−	6674.25i	0.710988	−	0.710988i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	5231.76			0.549893			0.274947	−	0.961460i	$-0.411340\pi$
$449$	5231.76			0.549893			0.274947	+	0.961460i	$0.411340\pi$
$450$		0			0
$451$	−3616.03	+	3616.03i	−0.377543	+	0.377543i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	265.978i		−	0.0274049i
$456$		0			0
$457$			6833.10i			0.699429i	0.936856	+	0.349715i	$0.113721\pi$
$457$			6833.10i			0.699429i	−0.936856	+	0.349715i	$0.886279\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5975.90	−	5975.90i	0.603742	−	0.603742i	−0.337561	−	0.941304i	$-0.609602\pi$
$461$	5975.90	−	5975.90i	0.603742	−	0.603742i	0.941304	+	0.337561i	$0.109602\pi$
$462$		0			0
$463$	4273.38			0.428943			0.214472	−	0.976730i	$-0.431197\pi$
$463$	4273.38			0.428943			0.214472	+	0.976730i	$0.431197\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12245.6	−	12245.6i	1.21340	−	1.21340i	0.243500	−	0.969901i	$-0.421704\pi$
$467$	12245.6	−	12245.6i	1.21340	−	1.21340i	0.969901	−	0.243500i	$-0.0782956\pi$
$468$		0			0
$469$	462.566	+	462.566i	0.0455422	+	0.0455422i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			10401.3i			1.01110i
$474$		0			0
$475$	−1822.28	−	1822.28i	−0.176025	−	0.176025i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4067.97			−0.388038			−0.194019	−	0.980998i	$-0.562152\pi$
$479$	−4067.97			−0.388038			−0.194019	+	0.980998i	$0.562152\pi$
$480$		0			0
$481$	−3269.06			−0.309888
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7030.06	+	7030.06i	0.658182	+	0.658182i
$486$		0			0
$487$			16174.3i			1.50499i	0.658600	+	0.752493i	$0.271149\pi$
$487$			16174.3i			1.50499i	−0.658600	+	0.752493i	$0.728851\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	13596.7	+	13596.7i	1.24971	+	1.24971i	0.955847	+	0.293866i	$0.0949420\pi$
$491$	13596.7	+	13596.7i	1.24971	+	1.24971i	0.293866	+	0.955847i	$0.405058\pi$
$492$		0			0
$493$	−9708.68	+	9708.68i	−0.886931	+	0.886931i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1192.61			0.107637
$498$		0			0
$499$	14646.7	−	14646.7i	1.31398	−	1.31398i	0.395530	−	0.918453i	$-0.370561\pi$
$499$	14646.7	−	14646.7i	1.31398	−	1.31398i	0.918453	−	0.395530i	$-0.129439\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	9828.84i		−	0.871265i	−0.900125	−	0.435632i	$-0.856525\pi$
$503$		−	9828.84i		−	0.871265i	0.900125	−	0.435632i	$-0.143475\pi$
$504$		0			0
$505$		−	5263.12i		−	0.463774i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−13456.1	+	13456.1i	−1.17177	+	1.17177i	−0.189985	+	0.981787i	$0.560844\pi$
$509$	−13456.1	+	13456.1i	−1.17177	+	1.17177i	−0.981787	+	0.189985i	$0.939156\pi$
$510$		0			0
$511$	−173.866			−0.0150516
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8351.51	+	8351.51i	−0.714585	+	0.714585i
$516$		0			0
$517$	1426.37	+	1426.37i	0.121338	+	0.121338i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			10607.1i			0.891950i	0.895045	+	0.445975i	$0.147143\pi$
$521$			10607.1i			0.891950i	−0.895045	+	0.445975i	$0.852857\pi$
$522$		0			0
$523$	−3903.15	−	3903.15i	−0.326334	−	0.326334i	0.524857	−	0.851191i	$-0.324119\pi$
$523$	−3903.15	−	3903.15i	−0.326334	−	0.326334i	−0.851191	+	0.524857i	$0.824119\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7113.36			0.587975
$528$		0			0
$529$	11536.0			0.948142
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1856.88	−	1856.88i	−0.150901	−	0.150901i
$534$		0			0
$535$		−	12643.3i		−	1.02172i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6194.39	+	6194.39i	0.495012	+	0.495012i
$540$		0			0
$541$	9532.77	−	9532.77i	0.757570	−	0.757570i	−0.218309	−	0.975880i	$-0.570054\pi$
$541$	9532.77	−	9532.77i	0.757570	−	0.757570i	0.975880	+	0.218309i	$0.0700541\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10298.5			0.809427
$546$		0			0
$547$	1232.88	−	1232.88i	0.0963693	−	0.0963693i	−0.657278	−	0.753648i	$-0.728292\pi$
$547$	1232.88	−	1232.88i	0.0963693	−	0.0963693i	0.753648	+	0.657278i	$0.228292\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			25710.6i			1.98786i
$552$		0			0
$553$		−	96.7105i		−	0.00743680i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2889.57	−	2889.57i	0.219812	−	0.219812i	−0.588607	−	0.808419i	$-0.700324\pi$
$557$	2889.57	−	2889.57i	0.219812	−	0.219812i	0.808419	+	0.588607i	$0.200324\pi$
$558$		0			0
$559$	−5341.19			−0.404129
$560$		0			0
$561$		0			0
$562$		0			0
$563$	70.0753	−	70.0753i	0.00524569	−	0.00524569i	−0.704479	−	0.709725i	$-0.748819\pi$
$563$	70.0753	−	70.0753i	0.00524569	−	0.00524569i	0.709725	+	0.704479i	$0.248819\pi$
$564$		0			0
$565$	3054.68	+	3054.68i	0.227453	+	0.227453i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	8915.23i		−	0.656847i	−0.944531	−	0.328423i	$-0.893483\pi$
$569$		−	8915.23i		−	0.656847i	0.944531	−	0.328423i	$-0.106517\pi$
$570$		0			0
$571$	−4946.30	−	4946.30i	−0.362515	−	0.362515i	0.502223	−	0.864738i	$-0.332516\pi$
$571$	−4946.30	−	4946.30i	−0.362515	−	0.362515i	−0.864738	+	0.502223i	$0.832516\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	644.818			0.0467665
$576$		0			0
$577$	17911.5			1.29232			0.646159	−	0.763203i	$-0.276374\pi$
$577$	17911.5			1.29232			0.646159	+	0.763203i	$0.276374\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	672.347	+	672.347i	0.0480097	+	0.0480097i
$582$		0			0
$583$			19140.4i			1.35972i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7940.26	+	7940.26i	0.558312	+	0.558312i	0.928827	−	0.370514i	$-0.120819\pi$
$587$	7940.26	+	7940.26i	0.558312	+	0.558312i	−0.370514	+	0.928827i	$0.620819\pi$
$588$		0			0
$589$	9418.83	−	9418.83i	0.658907	−	0.658907i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	7006.26			0.485181			0.242591	−	0.970129i	$-0.422003\pi$
$593$	7006.26			0.485181			0.242591	+	0.970129i	$0.422003\pi$
$594$		0			0
$595$	762.780	−	762.780i	0.0525562	−	0.0525562i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			8502.74i			0.579987i	0.957029	+	0.289994i	$0.0936532\pi$
$599$			8502.74i			0.579987i	−0.957029	+	0.289994i	$0.906347\pi$
$600$		0			0
$601$			11936.2i			0.810127i	0.914289	+	0.405063i	$0.132751\pi$
$601$			11936.2i			0.810127i	−0.914289	+	0.405063i	$0.867249\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5801.18	+	5801.18i	−0.389837	+	0.389837i
$606$		0			0
$607$	3850.00			0.257441			0.128721	−	0.991681i	$-0.458913\pi$
$607$	3850.00			0.257441			0.128721	+	0.991681i	$0.458913\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−732.459	+	732.459i	−0.0484977	+	0.0484977i
$612$		0			0
$613$	−6320.36	−	6320.36i	−0.416439	−	0.416439i	0.467536	−	0.883974i	$-0.345142\pi$
$613$	−6320.36	−	6320.36i	−0.416439	−	0.416439i	−0.883974	+	0.467536i	$0.845142\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	2585.09i		−	0.168674i	−0.996437	−	0.0843370i	$-0.973123\pi$
$617$		−	2585.09i		−	0.168674i	0.996437	−	0.0843370i	$-0.0268772\pi$
$618$		0			0
$619$	−7325.02	−	7325.02i	−0.475634	−	0.475634i	0.428098	−	0.903732i	$-0.359184\pi$
$619$	−7325.02	−	7325.02i	−0.475634	−	0.475634i	−0.903732	+	0.428098i	$0.859184\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1260.51			0.0810615
$624$		0			0
$625$	18174.8			1.16319
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−9375.13	−	9375.13i	−0.594294	−	0.594294i
$630$		0			0
$631$			14411.5i			0.909210i	0.890693	+	0.454605i	$0.150220\pi$
$631$			14411.5i			0.909210i	−0.890693	+	0.454605i	$0.849780\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−20534.3	−	20534.3i	−1.28327	−	1.28327i
$636$		0			0
$637$	−3180.91	+	3180.91i	−0.197853	+	0.197853i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	25724.0			1.58508			0.792542	−	0.609818i	$-0.208757\pi$
$641$	25724.0			1.58508			0.792542	+	0.609818i	$0.208757\pi$
$642$		0			0
$643$	7835.74	−	7835.74i	0.480578	−	0.480578i	−0.424738	−	0.905316i	$-0.639634\pi$
$643$	7835.74	−	7835.74i	0.480578	−	0.480578i	0.905316	+	0.424738i	$0.139634\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1247.43i			0.0757981i	0.999282	+	0.0378991i	$0.0120665\pi$
$647$			1247.43i			0.0757981i	−0.999282	+	0.0378991i	$0.987933\pi$
$648$		0			0
$649$			1693.89i			0.102451i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8302.21	+	8302.21i	−0.497535	+	0.497535i	−0.910670	−	0.413135i	$-0.864434\pi$
$653$	−8302.21	+	8302.21i	−0.497535	+	0.497535i	0.413135	+	0.910670i	$0.364434\pi$
$654$		0			0
$655$	−7003.63			−0.417793
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1696.16	−	1696.16i	0.100262	−	0.100262i	−0.655196	−	0.755459i	$-0.727414\pi$
$659$	1696.16	−	1696.16i	0.100262	−	0.100262i	0.755459	+	0.655196i	$0.227414\pi$
$660$		0			0
$661$	−8788.30	−	8788.30i	−0.517134	−	0.517134i	0.399569	−	0.916703i	$-0.369160\pi$
$661$	−8788.30	−	8788.30i	−0.517134	−	0.517134i	−0.916703	+	0.399569i	$0.869160\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	2020.00i		−	0.117793i
$666$		0			0
$667$	−4548.88	−	4548.88i	−0.264068	−	0.264068i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7037.56			0.404891
$672$		0			0
$673$	−23869.3			−1.36716			−0.683578	−	0.729878i	$-0.739577\pi$
$673$	−23869.3			−1.36716			−0.683578	+	0.729878i	$0.739577\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	5663.30	+	5663.30i	0.321504	+	0.321504i	0.849344	−	0.527840i	$-0.176998\pi$
$677$	5663.30	+	5663.30i	0.321504	+	0.321504i	−0.527840	+	0.849344i	$0.676998\pi$
$678$		0			0
$679$			1327.71i			0.0750410i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9152.80	−	9152.80i	−0.512770	−	0.512770i	0.402604	−	0.915374i	$-0.368105\pi$
$683$	−9152.80	−	9152.80i	−0.512770	−	0.512770i	−0.915374	+	0.402604i	$0.868105\pi$
$684$		0			0
$685$	−7437.11	+	7437.11i	−0.414828	+	0.414828i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−9828.85			−0.543468
$690$		0			0
$691$	−17057.9	+	17057.9i	−0.939091	+	0.939091i	−0.998249	−	0.0591580i	$-0.981158\pi$
$691$	−17057.9	+	17057.9i	−0.939091	+	0.939091i	0.0591580	+	0.998249i	$0.481158\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	29325.4i		−	1.60054i
$696$		0			0
$697$		−	10650.4i		−	0.578787i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	7720.44	−	7720.44i	0.415973	−	0.415973i	−0.467840	−	0.883813i	$-0.654968\pi$
$701$	7720.44	−	7720.44i	0.415973	−	0.415973i	0.883813	+	0.467840i	$0.154968\pi$
$702$		0			0
$703$	−24827.3			−1.33198
$704$		0			0
$705$		0			0
$706$		0			0
$707$	497.001	−	497.001i	0.0264380	−	0.0264380i
$708$		0			0
$709$	4577.66	+	4577.66i	0.242479	+	0.242479i	0.817875	−	0.575396i	$-0.195152\pi$
$709$	4577.66	+	4577.66i	0.242479	+	0.242479i	−0.575396	+	0.817875i	$0.695152\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			3332.88i			0.175059i
$714$		0			0
$715$	2953.40	+	2953.40i	0.154477	+	0.154477i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−30210.0			−1.56696			−0.783479	−	0.621418i	$-0.786557\pi$
$719$	−30210.0			−1.56696			−0.783479	+	0.621418i	$0.786557\pi$
$720$		0			0
$721$	−1577.28			−0.0814717
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4648.78	+	4648.78i	0.238140	+	0.238140i
$726$		0			0
$727$		−	20721.3i		−	1.05710i	−0.848903	−	0.528549i	$-0.822736\pi$
$727$		−	20721.3i		−	1.05710i	0.848903	−	0.528549i	$-0.177264\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−15317.6	−	15317.6i	−0.775025	−	0.775025i
$732$		0			0
$733$	−13879.8	+	13879.8i	−0.699404	+	0.699404i	−0.964282	−	0.264878i	$-0.914668\pi$
$733$	−13879.8	+	13879.8i	−0.699404	+	0.699404i	0.264878	+	0.964282i	$0.414668\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−10272.6			−0.513428
$738$		0			0
$739$	−8793.93	+	8793.93i	−0.437740	+	0.437740i	−0.891251	−	0.453511i	$-0.850171\pi$
$739$	−8793.93	+	8793.93i	−0.437740	+	0.437740i	0.453511	+	0.891251i	$0.350171\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			7669.27i			0.378678i	0.981912	+	0.189339i	$0.0606346\pi$
$743$			7669.27i			0.378678i	−0.981912	+	0.189339i	$0.939365\pi$
$744$		0			0
$745$		−	559.327i		−	0.0275062i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1193.92	−	1193.92i	0.0582443	−	0.0582443i
$750$		0			0
$751$	26531.8			1.28916			0.644580	−	0.764537i	$-0.277032\pi$
$751$	26531.8			1.28916			0.644580	+	0.764537i	$0.277032\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	9348.40	−	9348.40i	0.450627	−	0.450627i
$756$		0			0
$757$	−79.4192	−	79.4192i	−0.00381313	−	0.00381313i	0.705198	−	0.709011i	$-0.250858\pi$
$757$	−79.4192	−	79.4192i	−0.00381313	−	0.00381313i	−0.709011	+	0.705198i	$0.750858\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	36991.3i		−	1.76207i	−0.473055	−	0.881033i	$-0.656849\pi$
$761$		−	36991.3i		−	1.76207i	0.473055	−	0.881033i	$-0.343151\pi$
$762$		0			0
$763$	972.494	+	972.494i	0.0461424	+	0.0461424i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−869.835			−0.0409490
$768$		0			0
$769$	26637.0			1.24910			0.624548	−	0.780987i	$-0.285283\pi$
$769$	26637.0			1.24910			0.624548	+	0.780987i	$0.285283\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−19743.6	−	19743.6i	−0.918667	−	0.918667i	0.0782657	−	0.996933i	$-0.475062\pi$
$773$	−19743.6	−	19743.6i	−0.918667	−	0.918667i	−0.996933	+	0.0782657i	$0.975062\pi$
$774$		0			0
$775$		−	3406.07i		−	0.157871i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−14102.3	−	14102.3i	−0.648610	−	0.648610i
$780$		0			0
$781$	−13242.6	+	13242.6i	−0.606734	+	0.606734i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−33079.9			−1.50404
$786$		0			0
$787$	28878.1	−	28878.1i	1.30800	−	1.30800i	0.385136	−	0.922860i	$-0.374155\pi$
$787$	28878.1	−	28878.1i	1.30800	−	1.30800i	0.922860	−	0.385136i	$-0.125845\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			576.912i			0.0259325i
$792$		0			0
$793$			3613.88i			0.161832i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	16656.0	−	16656.0i	0.740257	−	0.740257i	−0.232371	−	0.972627i	$-0.574648\pi$
$797$	16656.0	−	16656.0i	0.740257	−	0.740257i	0.972627	+	0.232371i	$0.0746483\pi$
$798$		0			0
$799$	−4201.14			−0.186015
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1930.60	−	1930.60i	0.0848435	−	0.0848435i
$804$		0			0
$805$	357.391	+	357.391i	0.0156477	+	0.0156477i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			34940.4i			1.51847i	0.650819	+	0.759233i	$0.274426\pi$
$809$			34940.4i			1.51847i	−0.650819	+	0.759233i	$0.725574\pi$
$810$		0			0
$811$	15168.2	+	15168.2i	0.656753	+	0.656753i	0.954610	−	0.297857i	$-0.0962719\pi$
$811$	15168.2	+	15168.2i	0.656753	+	0.656753i	−0.297857	+	0.954610i	$0.596272\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	12322.0			0.529597
$816$		0			0
$817$	−40564.3			−1.73705
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−8710.55	−	8710.55i	−0.370280	−	0.370280i	0.497299	−	0.867579i	$-0.334325\pi$
$821$	−8710.55	−	8710.55i	−0.370280	−	0.370280i	−0.867579	+	0.497299i	$0.834325\pi$
$822$		0			0
$823$		−	24493.5i		−	1.03741i	−0.854952	−	0.518707i	$-0.826414\pi$
$823$		−	24493.5i		−	1.03741i	0.854952	−	0.518707i	$-0.173586\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	26328.0	+	26328.0i	1.10703	+	1.10703i	0.993539	+	0.113492i	$0.0362037\pi$
$827$	26328.0	+	26328.0i	1.10703	+	1.10703i	0.113492	+	0.993539i	$0.463796\pi$
$828$		0			0
$829$	9108.25	−	9108.25i	0.381596	−	0.381596i	−0.490081	−	0.871677i	$-0.663033\pi$
$829$	9108.25	−	9108.25i	0.381596	−	0.381596i	0.871677	+	0.490081i	$0.163033\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−18244.6			−0.758870
$834$		0			0
$835$	−30037.1	+	30037.1i	−1.24488	+	1.24488i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1394.89i			0.0573982i	0.999588	+	0.0286991i	$0.00913646\pi$
$839$			1394.89i			0.0573982i	−0.999588	+	0.0286991i	$0.990864\pi$
$840$		0			0
$841$		−	41200.9i		−	1.68932i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	17552.4	−	17552.4i	0.714583	−	0.714583i
$846$		0			0
$847$	−1095.62			−0.0444463
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4392.60	−	4392.60i	0.176941	−	0.176941i
$852$		0			0
$853$	15284.7	+	15284.7i	0.613527	+	0.613527i	0.943863	−	0.330337i	$-0.107162\pi$
$853$	15284.7	+	15284.7i	0.613527	+	0.613527i	−0.330337	+	0.943863i	$0.607162\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			2273.70i			0.0906277i	0.998973	+	0.0453139i	$0.0144288\pi$
$857$			2273.70i			0.0906277i	−0.998973	+	0.0453139i	$0.985571\pi$
$858$		0			0
$859$	21674.3	+	21674.3i	0.860905	+	0.860905i	0.991443	−	0.130538i	$-0.0416706\pi$
$859$	21674.3	+	21674.3i	0.860905	+	0.860905i	−0.130538	+	0.991443i	$0.541671\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	23721.7			0.935686			0.467843	−	0.883812i	$-0.345031\pi$
$863$	23721.7			0.935686			0.467843	+	0.883812i	$0.345031\pi$
$864$		0			0
$865$	−22628.5			−0.889469
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1073.87	+	1073.87i	0.0419200	+	0.0419200i
$870$		0			0
$871$		−	5275.12i		−	0.205213i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1413.26	+	1413.26i	0.0546020	+	0.0546020i
$876$		0			0
$877$	22429.6	−	22429.6i	0.863617	−	0.863617i	−0.128139	−	0.991756i	$-0.540900\pi$
$877$	22429.6	−	22429.6i	0.863617	−	0.863617i	0.991756	+	0.128139i	$0.0409004\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−24603.0			−0.940859			−0.470429	−	0.882438i	$-0.655901\pi$
$881$	−24603.0			−0.940859			−0.470429	+	0.882438i	$0.655901\pi$
$882$		0			0
$883$	23486.7	−	23486.7i	0.895120	−	0.895120i	−0.0998799	−	0.995000i	$-0.531846\pi$
$883$	23486.7	−	23486.7i	0.895120	−	0.895120i	0.995000	+	0.0998799i	$0.0318459\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	39722.9i		−	1.50368i	−0.659345	−	0.751841i	$-0.729166\pi$
$887$		−	39722.9i		−	1.50368i	0.659345	−	0.751841i	$-0.270834\pi$
$888$		0			0
$889$		−	3878.15i		−	0.146309i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−5562.75	+	5562.75i	−0.208455	+	0.208455i
$894$		0			0
$895$	18788.7			0.701716
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−24028.2	+	24028.2i	−0.891420	+	0.891420i
$900$		0			0
$901$	−28187.5	−	28187.5i	−1.04225	−	1.04225i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	51104.2i		−	1.87708i
$906$		0			0
$907$	−4565.44	−	4565.44i	−0.167136	−	0.167136i	0.618583	−	0.785719i	$-0.287707\pi$
$907$	−4565.44	−	4565.44i	−0.167136	−	0.167136i	−0.785719	+	0.618583i	$0.787707\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−2013.95			−0.0732438			−0.0366219	−	0.999329i	$-0.511660\pi$
$911$	−2013.95			−0.0732438			−0.0366219	+	0.999329i	$0.511660\pi$
$912$		0			0
$913$	−14931.4			−0.541245
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−661.359	−	661.359i	−0.0238168	−	0.0238168i
$918$		0			0
$919$		−	37746.5i		−	1.35489i	−0.735575	−	0.677443i	$-0.763088\pi$
$919$		−	37746.5i		−	1.35489i	0.735575	−	0.677443i	$-0.236912\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−6800.28	−	6800.28i	−0.242507	−	0.242507i
$924$		0			0
$925$	−4489.07	+	4489.07i	−0.159567	+	0.159567i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−45643.5			−1.61197			−0.805983	−	0.591939i	$-0.798363\pi$
$929$	−45643.5			−1.61197			−0.805983	+	0.591939i	$0.798363\pi$
$930$		0			0
$931$	−24157.8	+	24157.8i	−0.850418	+	0.850418i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			16939.7i			0.592501i
$936$		0			0
$937$			47317.5i			1.64973i	0.565331	+	0.824864i	$0.308748\pi$
$937$			47317.5i			1.64973i	−0.565331	+	0.824864i	$0.691252\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15074.7	+	15074.7i	−0.522234	+	0.522234i	−0.918246	−	0.396012i	$-0.870394\pi$
$941$	−15074.7	+	15074.7i	−0.522234	+	0.522234i	0.396012	+	0.918246i	$0.370394\pi$
$942$		0			0
$943$	4990.14			0.172324
$944$		0			0
$945$		0			0
$946$		0			0
$947$	14567.7	−	14567.7i	0.499880	−	0.499880i	−0.411521	−	0.911400i	$-0.635002\pi$
$947$	14567.7	−	14567.7i	0.499880	−	0.499880i	0.911400	+	0.411521i	$0.135002\pi$
$948$		0			0
$949$	991.388	+	991.388i	0.0339113	+	0.0339113i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			42987.2i			1.46117i	0.682824	+	0.730583i	$0.260752\pi$
$953$			42987.2i			1.46117i	−0.682824	+	0.730583i	$0.739248\pi$
$954$		0			0
$955$	3737.87	+	3737.87i	0.126654	+	0.126654i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1404.59			−0.0472956
$960$		0			0
$961$	−12186.0			−0.409049
$962$		0			0
$963$		0			0
$964$		0			0
$965$	19676.5	+	19676.5i	0.656381	+	0.656381i
$966$		0			0
$967$		−	44030.7i		−	1.46425i	−0.681170	−	0.732126i	$-0.738528\pi$
$967$		−	44030.7i		−	1.46425i	0.681170	−	0.732126i	$-0.261472\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	35699.8	+	35699.8i	1.17988	+	1.17988i	0.979775	+	0.200101i	$0.0641270\pi$
$971$	35699.8	+	35699.8i	1.17988	+	1.17988i	0.200101	+	0.979775i	$0.435873\pi$
$972$		0			0
$973$	2769.23	−	2769.23i	0.0912409	−	0.0912409i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	49515.3			1.62143			0.810714	−	0.585442i	$-0.199079\pi$
$977$	49515.3			1.62143			0.810714	+	0.585442i	$0.199079\pi$
$978$		0			0
$979$	−13996.6	+	13996.6i	−0.456930	+	0.456930i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	40046.2i		−	1.29936i	−0.760206	−	0.649682i	$-0.774902\pi$
$983$		−	40046.2i		−	1.29936i	0.760206	−	0.649682i	$-0.225098\pi$
$984$		0			0
$985$		−	18043.3i		−	0.583662i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7176.90	−	7176.90i	0.230750	−	0.230750i
$990$		0			0
$991$	−18673.2			−0.598560			−0.299280	−	0.954165i	$-0.596746\pi$
$991$	−18673.2			−0.598560			−0.299280	+	0.954165i	$0.596746\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−43306.3	+	43306.3i	−1.37980	+	1.37980i
$996$		0			0
$997$	21982.1	+	21982.1i	0.698274	+	0.698274i	0.964038	−	0.265764i	$-0.0856243\pi$
$997$	21982.1	+	21982.1i	0.698274	+	0.698274i	−0.265764	+	0.964038i	$0.585624\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.4.k.a.433.1		10
3.2	odd	2		64.4.e.a.49.5		10
4.3	odd	2		144.4.k.a.37.4		10
12.11	even	2		16.4.e.a.5.2	✓	10
16.3	odd	4		144.4.k.a.109.4		10
16.13	even	4	inner	576.4.k.a.145.1		10
24.5	odd	2		128.4.e.a.97.1		10
24.11	even	2		128.4.e.b.97.5		10
48.5	odd	4		128.4.e.a.33.1		10
48.11	even	4		128.4.e.b.33.5		10
48.29	odd	4		64.4.e.a.17.5		10
48.35	even	4		16.4.e.a.13.2	yes	10
96.5	odd	8		1024.4.b.k.513.10		10
96.11	even	8		1024.4.b.j.513.10		10
96.29	odd	8		1024.4.a.m.1.10		10
96.35	even	8		1024.4.a.n.1.1		10
96.53	odd	8		1024.4.b.k.513.1		10
96.59	even	8		1024.4.b.j.513.1		10
96.77	odd	8		1024.4.a.m.1.1		10
96.83	even	8		1024.4.a.n.1.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.2	✓	10	12.11	even	2
16.4.e.a.13.2	yes	10	48.35	even	4
64.4.e.a.17.5		10	48.29	odd	4
64.4.e.a.49.5		10	3.2	odd	2
128.4.e.a.33.1		10	48.5	odd	4
128.4.e.a.97.1		10	24.5	odd	2
128.4.e.b.33.5		10	48.11	even	4
128.4.e.b.97.5		10	24.11	even	2
144.4.k.a.37.4		10	4.3	odd	2
144.4.k.a.109.4		10	16.3	odd	4
576.4.k.a.145.1		10	16.13	even	4	inner
576.4.k.a.433.1		10	1.1	even	1	trivial
1024.4.a.m.1.1		10	96.77	odd	8
1024.4.a.m.1.10		10	96.29	odd	8
1024.4.a.n.1.1		10	96.35	even	8
1024.4.a.n.1.10		10	96.83	even	8
1024.4.b.j.513.1		10	96.59	even	8
1024.4.b.j.513.10		10	96.11	even	8
1024.4.b.k.513.1		10	96.53	odd	8
1024.4.b.k.513.10		10	96.5	odd	8

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	576.k (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-8.67959 - 8.67959i) q^{5} -1.63924i q^{7} +O(q^{10})\)	\(q+(-8.67959 - 8.67959i) q^{5} -1.63924i q^{7} +(18.2021 + 18.2021i) q^{11} +(-9.34700 + 9.34700i) q^{13} -53.6113 q^{17} +(-70.9870 + 70.9870i) q^{19} -25.1189i q^{23} +25.6706i q^{25} +(181.094 - 181.094i) q^{29} -132.684 q^{31} +(-14.2280 + 14.2280i) q^{35} +(174.872 + 174.872i) q^{37} +198.660i q^{41} +(285.717 + 285.717i) q^{43} +78.3629 q^{47} +340.313 q^{49} +(525.776 + 525.776i) q^{53} -315.973i q^{55} +(46.5301 + 46.5301i) q^{59} +(193.318 - 193.318i) q^{61} +162.256 q^{65} +(-282.182 + 282.182i) q^{67} +727.536i q^{71} -106.065i q^{73} +(29.8376 - 29.8376i) q^{77} +58.9970 q^{79} +(-410.156 + 410.156i) q^{83} +(465.324 + 465.324i) q^{85} +768.959i q^{89} +(15.3220 + 15.3220i) q^{91} +1232.28 q^{95} -809.953 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{5}+O(q^{10}) \) 10 * q + 2 * q^5	\( 10 q + 2 q^{5} + 18 q^{11} - 2 q^{13} + 4 q^{17} + 26 q^{19} + 202 q^{29} - 368 q^{31} + 476 q^{35} - 10 q^{37} + 838 q^{43} - 944 q^{47} + 94 q^{49} + 378 q^{53} + 1706 q^{59} + 910 q^{61} + 492 q^{65} - 1942 q^{67} + 268 q^{77} + 4416 q^{79} - 2562 q^{83} - 12 q^{85} - 3332 q^{91} + 6900 q^{95} - 4 q^{97}+O(q^{100}) \) 10 * q + 2 * q^5 + 18 * q^11 - 2 * q^13 + 4 * q^17 + 26 * q^19 + 202 * q^29 - 368 * q^31 + 476 * q^35 - 10 * q^37 + 838 * q^43 - 944 * q^47 + 94 * q^49 + 378 * q^53 + 1706 * q^59 + 910 * q^61 + 492 * q^65 - 1942 * q^67 + 268 * q^77 + 4416 * q^79 - 2562 * q^83 - 12 * q^85 - 3332 * q^91 + 6900 * q^95 - 4 * q^97

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